Properties

Label 2-1050-105.104-c1-0-12
Degree $2$
Conductor $1050$
Sign $-0.241 - 0.970i$
Analytic cond. $8.38429$
Root an. cond. $2.89556$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + (−0.618 + 1.61i)3-s + 4-s + (−0.618 + 1.61i)6-s + (−0.381 − 2.61i)7-s + 8-s + (−2.23 − 2.00i)9-s + 4.47i·11-s + (−0.618 + 1.61i)12-s − 1.23·13-s + (−0.381 − 2.61i)14-s + 16-s + 5.23i·17-s + (−2.23 − 2.00i)18-s + 8.47i·19-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.356 + 0.934i)3-s + 0.5·4-s + (−0.252 + 0.660i)6-s + (−0.144 − 0.989i)7-s + 0.353·8-s + (−0.745 − 0.666i)9-s + 1.34i·11-s + (−0.178 + 0.467i)12-s − 0.342·13-s + (−0.102 − 0.699i)14-s + 0.250·16-s + 1.26i·17-s + (−0.527 − 0.471i)18-s + 1.94i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.241 - 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.241 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.241 - 0.970i$
Analytic conductor: \(8.38429\)
Root analytic conductor: \(2.89556\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1050} (1049, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1/2),\ -0.241 - 0.970i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.852522346\)
\(L(\frac12)\) \(\approx\) \(1.852522346\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + (0.618 - 1.61i)T \)
5 \( 1 \)
7 \( 1 + (0.381 + 2.61i)T \)
good11 \( 1 - 4.47iT - 11T^{2} \)
13 \( 1 + 1.23T + 13T^{2} \)
17 \( 1 - 5.23iT - 17T^{2} \)
19 \( 1 - 8.47iT - 19T^{2} \)
23 \( 1 - 4T + 23T^{2} \)
29 \( 1 - 7.70iT - 29T^{2} \)
31 \( 1 - 2.76iT - 31T^{2} \)
37 \( 1 + 0.763iT - 37T^{2} \)
41 \( 1 - 2.47T + 41T^{2} \)
43 \( 1 + 4.94iT - 43T^{2} \)
47 \( 1 + 6.47iT - 47T^{2} \)
53 \( 1 + 0.472T + 53T^{2} \)
59 \( 1 - 4.47T + 59T^{2} \)
61 \( 1 + 7.23iT - 61T^{2} \)
67 \( 1 - 12iT - 67T^{2} \)
71 \( 1 + 7.23iT - 71T^{2} \)
73 \( 1 + 11.2T + 73T^{2} \)
79 \( 1 - 8.94T + 79T^{2} \)
83 \( 1 + 14.6iT - 83T^{2} \)
89 \( 1 + 5.52T + 89T^{2} \)
97 \( 1 + 0.763T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.43818702649946889995170431497, −9.662706753134410956310718751659, −8.521067081240456403604668978034, −7.43433655877233605711032841261, −6.68441058385257951394634051310, −5.68336357185931872345313926735, −4.81262740963840833702965431464, −4.02328040981591173493777339452, −3.36536578159568296727130980861, −1.67404563871674716494633397110, 0.69110077047701441249159540905, 2.52524051686233231605210669586, 2.92452096857446816300009630787, 4.69365316308581384196775364594, 5.46030595039366272031243137357, 6.19742712720476688517660351696, 6.95072063394678907984108244673, 7.84343890121371246165101792915, 8.784866465680478567129044671031, 9.524808744883528074728109374798

Graph of the $Z$-function along the critical line